Results 1  10
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33
Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
 J. Amer. Math. Soc
, 1996
"... Recall that a subset of R n is called semialgebraic if it can be represented as a (finite) boolean combination of sets of the form {�α ∈ R n: p(�α) =0}, {�α ∈R n: q(�α)>0}where p(�x), q(�x) arenvariable polynomials with real coefficients. A map from R n to R m is called semialgebraic if its gr ..."
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Cited by 107 (4 self)
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Recall that a subset of R n is called semialgebraic if it can be represented as a (finite) boolean combination of sets of the form {�α ∈ R n: p(�α) =0}, {�α ∈R n: q(�α)>0}where p(�x), q(�x) arenvariable polynomials with real coefficients. A map from R n to R m is called semialgebraic if its graph, considered
Robust Geometric Computation
, 1997
"... Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section... ..."
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Cited by 73 (11 self)
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Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section...
Efficient solving of quantified inequality constraints over the real numbers
 ACM Transactions on Computational Logic
"... Let a quantified inequality constraint over the reals be a formula in the firstorder predicate language over the structure of the real numbers, where the allowed predicate symbols are ≤ and <. Solving such constraints is an undecidable problem when allowing function symbols such sin or cos. In t ..."
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Cited by 33 (8 self)
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Let a quantified inequality constraint over the reals be a formula in the firstorder predicate language over the structure of the real numbers, where the allowed predicate symbols are ≤ and <. Solving such constraints is an undecidable problem when allowing function symbols such sin or cos. In the paper we give an algorithm that terminates with a solution for all, except for very special, pathological inputs. We ensure the practical efficiency of this algorithm by employing constraint programming techniques. 1
Continuous FirstOrder Constraint Satisfaction
 ARTIFICIAL INTELLIGENCE, AUTOMATED REASONING, AND SYMBOLIC COMPUTATION, NUMBER 2385 IN LNCS
, 2002
"... This paper shows how to use constraint programming techniques for solving firstorder constraints over the reals (i.e., formulas in the firstorder predicate language over the structure of the real numbers). More specifically, based on a narrowing operator that implements an arbitrary notion of con ..."
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Cited by 28 (12 self)
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This paper shows how to use constraint programming techniques for solving firstorder constraints over the reals (i.e., formulas in the firstorder predicate language over the structure of the real numbers). More specifically, based on a narrowing operator that implements an arbitrary notion of consistency for atomic constraints over the reals (e.g., boxconsistency), the paper provides a narrowing operator for firstorder constraints that implements a corresponding notion of firstorder consistency, and a solver based on such a narrowing operator. As a consequence, this solver can take over various favorable properties from the field of constraint programming.
On the expressiveness and decidability of ominimal hybrid systems
 Journal of Complexity
"... Abstract. This paper is driven by a general motto: bisimulate a hybrid system by a finite symbolic dynamical system. In the case of ominimal hybrid systems, the continuous and discrete components can be decoupled, and hence, the problem reduces in building a finite symbolic dynamical system for the ..."
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Cited by 13 (6 self)
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Abstract. This paper is driven by a general motto: bisimulate a hybrid system by a finite symbolic dynamical system. In the case of ominimal hybrid systems, the continuous and discrete components can be decoupled, and hence, the problem reduces in building a finite symbolic dynamical system for the continuous dynamics of each location. We show that this can be done for a quite general class of hybrid systems defined on ominimal structures. In particular, we recover the main result of a paper by Lafferriere G., Pappas G.J. and Sastry S. on ominimal hybrid systems. We also study related decidability questions. Mathematics Subject Classification: 68Q60, 03C64, 03D15. 1
Quantifier Elimination for Neocompact Sets
 JOURNAL OF SYMBOLIC LOGIC
"... We shall prove quantifier elimination theorems for neocompact formulas, which define neocompact sets and are built from atomic formulas using finite disjunctions, infinite conjunctions, existential quantifiers, and bounded universal quantifiers. The neocompact sets were first introduced to provide a ..."
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Cited by 11 (9 self)
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We shall prove quantifier elimination theorems for neocompact formulas, which define neocompact sets and are built from atomic formulas using finite disjunctions, infinite conjunctions, existential quantifiers, and bounded universal quantifiers. The neocompact sets were first introduced to provide an easy alternative to nonstandard methods of proving existence theorems in probability theory, where they behave like compact sets. The quantifier elimination theorems in this paper can be applied in a general setting to show that the family of neocompact sets is countably compact. To provide the necessary setting we introduce the notion of a law structure. This notion was motivated by the probability law of a random variable. However, in this paper we discuss a variety of model theoretic examples of the notion in the light of our quantifier elimination results.
What is a ClosedForm Number?
 Amer. Math. Monthly
, 1999
"... this paper is to eliminate this scandal by suggesting a precise definition of a "closedform expression for a number." This will enable us to restate Questions 1 and 2 precisely, and will let us see how they are related to existing work in logic, computer algebra, and transcendental number ..."
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Cited by 11 (0 self)
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this paper is to eliminate this scandal by suggesting a precise definition of a "closedform expression for a number." This will enable us to restate Questions 1 and 2 precisely, and will let us see how they are related to existing work in logic, computer algebra, and transcendental number theory. My hope is that this definition of a closedform expression for a number will become standard, and that many readers will be lured into working on the many attractive open problems in this area. 2. From elementary functions to EL numbers
Model theory and exponentiation
 Notices Amer. Math. Soc
, 1996
"... Model theory is a branch of mathematical logic in which one studies mathematical structures by considering the firstorder sentences true of those structures and the sets definable in those structures by firstorder formulas. The fields of real and complex numbers have long served as motivating exam ..."
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Cited by 6 (0 self)
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Model theory is a branch of mathematical logic in which one studies mathematical structures by considering the firstorder sentences true of those structures and the sets definable in those structures by firstorder formulas. The fields of real and complex numbers have long served as motivating examples for model theorists. Many model theoretic concepts arose by abstracting classical algebraic phenomena to a more general setting (see, for example, [29]). In the past five years significant progress has been made in the other direction. Model theoretic methods have been used to develop new insights into real analytic geometry. In particular, this has led to a generalization of semialgebraic and subanalytic geometry to a setting in which one studies global exponentiation on the reals. Tarski’s Theorem The logical study of the field of real numbers began with the work of Tarski. These results were announced by Tarski in [33], but publication was interrupted by the war, and the proofs did not appear until [34]. Tarski was primarily David Marker is professor of mathematics at the University of Illinois at Chicago. His email address is
Noetherian Varieties in Definably Complete Structures
 the Journal of Logic and Analysis
"... We prove that the zeroset of a C ∞ function belonging to a noetherian differential ring M can be written as a finite union of C ∞ manifolds which are definable by functions from the same ring. These manifolds can be taken to be connected under the additional assumption that every zerodimensional r ..."
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Cited by 4 (4 self)
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We prove that the zeroset of a C ∞ function belonging to a noetherian differential ring M can be written as a finite union of C ∞ manifolds which are definable by functions from the same ring. These manifolds can be taken to be connected under the additional assumption that every zerodimensional regular zeroset of functions in M consists of finitely many points. These results hold not only for C ∞ functions over the reals, but more generally for definable C ∞ functions in a definably complete expansion of an ordered field. The class of definably complete expansions of ordered fields, whose basic properties are discussed in this paper, expands the class of real closed fields and includes ominimal expansions of ordered fields. Finally, we provide examples of noetherian differential rings of C ∞ functions over the reals, containing nonanalytic functions. 1
Inclusion Dynamics Hybrid Automata
, 2008
"... Hybrid systems are dynamical systems with the ability to describe mixed discretecontinuous evolution of a wide range of systems. Consequently, at first glance, hybrid systems appear powerful but recalcitrant, neither yielding to analysis and reasoning through a purely continuoustime modeling as wit ..."
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Cited by 4 (4 self)
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Hybrid systems are dynamical systems with the ability to describe mixed discretecontinuous evolution of a wide range of systems. Consequently, at first glance, hybrid systems appear powerful but recalcitrant, neither yielding to analysis and reasoning through a purely continuoustime modeling as with systems of differential equations, nor open to inferential processes commonly used for discrete statetransition systems such as finite state automata. A convenient and popular model, called hybrid automata, was introduced to model them and has spurred much interest on its tractability as a tool for inference and model checking in a general setting. Intuitively, a hybrid automaton is simply a “finitestate ” automaton with each state augmented by continuous variables, which evolve according to a set of welldefined continuous laws, each specified separately for each state. This article investigates both the notion of hybrid automaton and the model checking problem over such structure. In particular, it relates firstorder theories and analysis results on multivalued maps and reduces the bounded reachability problem for hybrid automata whose continuous laws are expressed by inclusions (x ′ ∈ f(x, t)) to a decidability problem for firstorder formulæ over the reals. Furthermore, the paper introduces a class of hybrid automata for which the reachability problem can be decided and shows that the problem of deciding whether a hybrid automaton belongs to this class can be again decided using firstorder formulæ over the reals. Despite the fact that the bisimulation quotient for this class of hybrid automata can be infinite, we show that our techniques permit effective model checking for a nontrivial fragment of CTL.